Infinitesimal rotations
SO(n) is for each n'' a Lie group. It is compact and connected, but not simply connected. It is also a semi-simple group, in fact a simple group with the exception SO(4). The Lie algebra of is denoted by and consists of all skew-symmetric matrices. Proposition 3.24 (The vector cross product can be expressed as the product of a skew-symmetric matrix and a vector). The most common basis for is : L_{\mathbf{x}} = e_3 \wedge e_2 = \left\begin{matrix}0&0&0\\0&0&-1\\0&1&0\end{matrix}\right : L_{\mathbf{y}} = e_1 \wedge e_3 = \left\begin{matrix}0&0&1\\0&0&0\\-1&0&0\end{matrix}\right : L_{\mathbf{z}} = e_2 \wedge e_1 = \left\begin{matrix}0&-1&0\\1&0&0\\0&0&0\end{matrix}\right The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives. If f(\theta_0) = I then f'(\theta_0) is in . Informally, an element of is the difference between the matrix of an infinitesimal rotation and the identity matrix, but "scaled up by a factor of infinity".Henning Makholm (https://math.stackexchange.com/users/14366/henning-makholm), If so(3) is the Lie algebra of SO(3) then why are the matrices of so(3) not rotation matrices?, URL (version: 2017-10-24): https://math.stackexchange.com/q/2488191 An actual "differential rotation", or ''infinitesimal rotation matrix has the form : I + A \, d\theta ~, where is vanishingly small and . These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals . To understand what this means, one considers : dA_{\bold{x}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}~ . (In 3 dimensions the trace of any rotation matrix must equal therefore the angle of rotation must be infinitesimal) First, test the orthogonality condition, I''}}. The product is : dA_{\bold{x}}^T \, dA_{\bold{x}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1+d\theta^2 & 0 \\ 0 & 0 & 1+d\theta^2 \end{bmatrix} , differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix. Next, examine the square of the matrix, : dA_{\bold{x}}^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1-d\theta^2 & -2d\theta \\ 0 & 2d\theta & 1-d\theta^2 \end{bmatrix}~. Again discarding second order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation, : dA_{\bold{y}} = \begin{bmatrix} 1 & 0 & d\phi \\ 0 & 1 & 0 \\ -d\phi & 0 & 1 \end{bmatrix} . Compare the products to , : \begin{align} dA_{\bold{x}}\,dA_{\bold{y}} &{}= \begin{bmatrix} 1 & 0 & d\phi \\ d\theta\,d\phi & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix} \\ dA_{\bold{y}}\,dA_{\bold{x}} &{}= \begin{bmatrix} 1 & d\theta\,d\phi & d\phi \\ 0 & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix}. \\ \end{align} Since is second order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is commutative. In fact, : dA_{\bold{x}}\,dA_{\bold{y}} = dA_{\bold{y}}\,dA_{\bold{x}} , \,\! again to first order. In other words, '''the order in which infinitesimal rotations are applied is irrelevant. This useful fact makes, for example, derivation of rigid body rotation relatively simple. But one must always be careful to distinguish (the first order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements. When contrasting the behavior of finite rotation matrices in the BCH formula above with that of infinitesimal rotation matrices, where all the commutator terms will be second order infinitesimals one finds a bona fide vector space. Technically, this dismissal of any second order terms amounts to Group contraction. Curl More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space V'' with an inner product may be defined as the bivectors on the space, which are sums of simple bivectors (2-blades) v \wedge w . The correspondence is given by the map v \wedge w \mapsto v^* \otimes w - w^* \otimes v, where v^* is the covector dual to the vector v ; in orthonormal coordinates these are exactly the elementary skew-symmetric matrices. This characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. The vector calculus operations of grad, curl, and div are most easily generalized and understood in the context of differential forms, which involves a number of steps. In a nutshell, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and , these all being 3-dimensional spaces. Lie algebra '''R'3 :See also Cross product Associated with every Lie group is its Lie algebra, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the Lie bracket. The Lie algebra of is denoted by and consists of all skew-symmetric matrices. Proposition 3.24 This may be seen by differentiating the orthogonality condition, I'', ''A ∈ SO(3)}}.For an alternative derivation of , see Classical group. The Lie bracket of two elements of is, as for the Lie algebra of every matrix group, given by the matrix commutator, ''A''1''A''2 − ''A''2''A''1}}, which is again a skew-symmetric matrix. The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the Baker–Campbell–Hausdorff formula. The elements of are the "infinitesimal generators" of rotations, i.e. they are the elements of the tangent space of the manifold SO(3) at the identity element. If ''R(φ, 'n') denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector n''', then : \left.{\operatorname{d}\over\operatorname{d}\phi} \right|_{\phi=0} R(\phi,\boldsymbol{n}) \boldsymbol{x} = \boldsymbol{n} \times \boldsymbol{x} for every vector '''x in R'3. This can be used to show that the Lie algebra (with commutator) is isomorphic to the Lie algebra '''R'3 (with cross product). Under this isomorphism, an Euler vector \boldsymbol{\omega}\in\mathbb R^3 corresponds to the linear map \bold{\tilde\omega} defined by \bold{\tilde\omega}(\boldsymbol{x})=\boldsymbol{\omega}\times\boldsymbol{x} . In more detail, a most often suitable basis for as a -dimensional}} vector space is : L_{\bold{x}} = \begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix} , \quad L_{\bold{y}} = \begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix} , \quad L_{\bold{z}} = \begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}. The commutation relations of these basis elements are, : L_{\bold{y}} = L_{\bold{z}}, \quad L_{\bold{x}} = L_{\bold{y}}, \quad L_{\bold{z}} = L_{\bold{x}} which agree with the relations of the three standard unit vectors of 'R'3 under the cross product. Cross product The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector: : \mathbf{a} \times \mathbf{b} = \mathbf{a}_{\times} \mathbf{b} = \begin{bmatrix}\,0&\!-a_3&\,\,a_2\\ \,\,a_3&0&\!-a_1\\-a_2&\,\,a_1&\,0\end{bmatrix}\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix} where '''a× is defined by: : \mathbf{a}_{\times} \stackrel{\rm def}{=} \begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix}. One actually has : (\mathbf{a\times b})_{\times} = \mathbf{a}_{\times}\mathbf{b}_{\times} - \mathbf{b}_{\times}\mathbf{a}_{\times} = \mathbf{a}_{\times},\mathbf{b}_{\times} i.e., the commutator of skew-symmetric three-by-three matrices can be identified with the cross-product of three-vectors. Since the skew-symmetric three-by-three matrices are the Lie algebra of the rotation group SO(3) this elucidates the relation between three-space \mathbb{R}^3 , the cross product and three-dimensional rotations. Infinitesimal rotations :See also Clifford algebra#Geometric product as higher dimensional equivalent of Complex numbers Skew-symmetric matrices over the field of real numbers form the tangent space to the real orthogonal group O(n'') at the identity matrix; formally, the special orthogonal Lie algebra. In this sense, then, skew-symmetric matrices can be thought of as infinitesimal rotations. Another way of saying this is that the space of skew-symmetric matrices forms the Lie algebra o(''n) of the Lie group O(n''). The Lie bracket on this space is given by the commutator: : A,B = AB - BA.\, It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric: : \begin{align} {}A,B{}^{\mathsf{T}} & = B^{\mathsf{T}}A^{\mathsf{T}}-A^{\mathsf{T}} B^{\mathsf{T}} \\ & = (-B)(-A) - (-A)(-B) = BA-AB = -A,B \, . \end{align} The matrix exponential of a skew-symmetric matrix ''A is then an orthogonal matrix R'': : R=\exp(A)=\sum_{n=0}^\infty \frac{A^n}{n!}. The image of the exponential map of a Lie algebra always lies in the connected component of the Lie group that contains the identity element. In the case of the Lie group O(''n), this connected component is the special orthogonal group SO(n''), consisting of all orthogonal matrices with determinant 1. So ''R = exp(A'') will have determinant +1. Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that ''every orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix. In the particular important case of dimension n'' = 2, the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus. Indeed, if n=2, a special orthogonal matrix has the form : \begin{bmatrix} a & -b \\ b & \,a \end{bmatrix}, with ''a''2 + ''b''2 = 1. Therefore, putting ''a = cos''θ'' and b'' = sin ''θ, it can be written : \begin{bmatrix} \cos\,\theta & -\sin\,\theta \\ \sin\,\theta & \,\cos\,\theta \end{bmatrix}= \exp\left( \theta \begin{bmatrix} 0 & -1 \\ 1 &\,0 \end{bmatrix} \right), which corresponds exactly to the polar form cos θ'' + ''i''sin ''θ = e''iθ'' of a complex number of unit modulus. The exponential representation of an orthogonal matrix of order n'' can also be obtained starting from the fact that in dimension ''n any special orthogonal matrix R'' can be written as ''R = QSQ''T, where ''Q is orthogonal and S is a block diagonal matrix with \scriptstyle\lfloor {n/2}\rfloor blocks of order 2, plus one of order 1 if n'' is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix ''S writes as exponential of a skew-symmetric block matrix Σ of the form above, S'' = exp(Σ), so that ''R = Q'' exp(Σ)''Q''T = exp(''QΣ''Q''T), exponential of the skew-symmetric matrix QΣ''Q''T. Conversely, the surjectivity of the exponential map, together with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices. References This article incorporates text from Wikipedia:Rotation matrix, Wikipedia:Skew-symmetric matrix, Wikipedia:Curl (mathematics)#Generalizations, Wikipedia:Rotation group SO(3), and stack exchange See also *Infinitesimal transformation Category:Geometry